1.1: Basic numbers on climate

Of a type first done by Svante Arrhenius in 1896, who got the modern numbers basically rightLet’s start with crude “back of the envelope” approximations and see how far we get towards understanding global warming. We’ll quickly hit a complexity barrier to this approach, but the lead-up was interesting to me.

We can start with radiation balance. This is basically the sum of the incoming radiation, sunlight, and the outgoing thermal radiation. The higher this is the more energy is stored in the atmosphere and by and large the hotter things will be.

An ideal black body radiator (e.g., a very idealized Planet Earth), with no greenhouse effect, would radiate an energy flux:

$σ = 5.67*10^{-8}Watts/meter^{2}/Kelvin^{4}$
This is the Stefan-Boltzmann constant, and T is the temperature in Kelvin
$no\space greenhouse\space outgoing\space flux = σ*T^{4}$

If we multiply this flux by the surface area of the Earth, we have the outgoing emitted energy:

$no\space greenhouse\space outgoing\space flux = 4 * π * (radius\space of\space earth)^{2}* σ*T^{4}$

If the planet was in radiative balance — so that the incoming solar energy is equal to the outgoing thermal energy and the planet would be neither warming nor cooling — we’d want to set this equal to the incoming solar energy

$incoming \space solar \space energy = S * (1-A) * π * (radius\space of\space earth)^{2}$

This last equation introduced the albedo A, which is the fraction of incoming solar energy reflected directly back without being absorbed, and the incoming solar flux S from the sun.

The incoming solar radiation is roughly S = 1.361 kilowatts per square meter, and the albedo A is around 0.3. S is easy to calculate: the sun has a radius of 700 million meters and a temperature of 5,778K, so the energy radiated by the sun is, by the same formula we used for the Earth,

$σ*4 * π * (700\space million\space meters)^{2}* (5778\space Kelvin)^{4}$

which is then by the time it reaches the Earth spread over a sphere of radius given by the Sun-Earth distance of 1.5e11 meters, so the solar flux per unit area is

(That’s a lot of solar flux. If the area of Texas was covered by 4% efficient solar collection, that would offset the entire world’s average energy consumption.)$S = \dfrac{ σ*4 * π * (700\space million\space meters)^{2}* (5778\space Kelvin)^{4}}{4*π* (1.5e11\space meters)^{2}} = 1.376 kW/meter^{2}$

We can then solve our incoming-outgoing solar energy balance for the single unknown, the equilibrium temperature T, which comes out to 255 Kelvin or so. Now this is quite cold, below zero degrees Fahrenheit, whereas the actual global mean surface temperature is much warmer that, on average around 60 degrees Fahrenheit (288K or so) — the reason for this difference is the greenhouse effect.

The greenhouse-gas-warmed situation that we find ourselves in is also simple to model at a crude level, but complex to model at a more accurate level.

e.g., by its energy being converted into intra-molecular vibrations and rotations in the CO2 molecules in the atmosphere

A more complete explanation is in the lead-up to equation 1.15 here, or here
At the crudest level, if a certain fraction ε of outgoing infrared radiation is absorbed by the atmosphere on the way out and then re-emitted, then we can calculate the resulting ground temperature

f is the fraction of incoming solar radiation absorbed by the atmosphere before making it to the surface. I’ve seen f = 0.08 (the atmosphere absorbs very little, less than 10%, of the incoming solar radiation

ε is the so-called atmospheric emissivity. It is zero for no greenhouse gasses and 1 for complete absorption of infrared radiation from the surface by the atmosphere. I've also seen and ε=0.89 although various other combinations do the same job. Even if ε=1, we don’t have infinite warming, as the radiation is eventually re-radiated out to space
$T = [\dfrac{(2-f)*(1-A)*S}{(2-ε)*4*σ}]^{1/4}$

If you plug the values of f and ε from the margin note in you get a ground temperature of 286 Kelvin or so, which is pretty close to our actual global mean surface temperature of around 288 Kelvin

Again, this is a very limited model, because it neglects almost all the complexities of the atmosphere and climate system, but it can be used to illustrate certain basic things. For example, if you want to keep the Earth’s temperature low, you can theoretically increase the albedo, A, to reflect a higher percentage of the incoming solar radiation back to space; this is called solar geoengineering or solar radiation management, and this formula allows to calculate crudely the kind of impact that can have.